Entropy production is a central quantity to characterize non-equilibrium mesoscopic systems. Recently, new (and surprising) generic properties of entropy production have been discovered. It is unclear if there are even more generic properties of entropy production, and how these properties are related. In this talk, I will present a general theory for non-equilibrium physical systems described by overdamped Langevin equations. For these system, entropy production evolves according to a simple stochastic differential equation. At steady state, a random time transformation maps this evolution into a model-independent form. This implies several generic properties for the entropy production, such as a finite-time uncertainty equality, universal distributions of the infimum and the supremum before the infimum, and universal distribution of the number of zero-crossings. In the second part of my talk I will discuss how the arcsine law, a general result in the theory of Brownian motion, applies to currents in stochastic thermodynamics.

Due to the advance of single-molecule techniques, stochastic modeling and computation become more and more useful and popular recently. I will discuss two related theories. One is a unifying mathematical theory of nonequilibrium thermodynamics of chemical reaction systems. A generalized macroscopic free energy called landscape emerges and satisfies a balance equation. The balance equation is valid generally in isothermal driven systems, which is actually an unknown form of the second law. This framework is totally based on the internal kinetics of the system, without knowing every details of the interaction between internal kinetics that we can measure in most experiments with all the surroundings. The other is the landscape theory and a new rate formula for the phenotype transition in an intermediate scenario of a single cell, which is more general and more close to the reality of living cells. The new rate formula can explain a "noise enhancer" therapy for HIV reported recently, which motivated a future project of us.

Speaker

Dr. Raphael Chetrite (CNRS, France)

Time

November 13 (Mon) 15:00-16:00

Place

工学部6号館 3F セミナー室B

Title

On Gibbs-Shannon Entropy

Abstract

This talk will be focus on the question of the physical contents of the Gibbs-Shannon entropy outside equilibrium. Article : Gavrilov-Chetrite-Bechhoeffer : Direct measurement of weakly nonequilibrium system entropy is consistent with Gibbs-Shannon. PNAS (2017)

Efficiency and power (extracted work per unit time) are two important quantities to characterize heat engines. It has been believed that these two quantities are complementary, i.e., an engine with high efficiency inevitably works slowly. However, this belief has not yet been proven, and even worse, maybe surprisingly, whether finite power and the maximum efficiency (Carnot efficiency) are compatible has still been an open problem. We note that conventional thermodynamics does not prohibit the compatibility because speed of the opera tion is out of the scope of thermodynamics, and linear irreversible thermodynamics neither prohibit even in the linear response regime if time-reversal symmetry is broken [1]. Triggered by the latter work, many works have investigated the relation between finite power and the Carnot efficiency on the basis of specific models mostly within the linear response regime [2-9]. Although some papers have proposed abstract ideas for the coexistence [2-4], all analyses on concrete models in the linear response regime have shown that finite power and the Carnot efficiency are incompatible within these models [5-9]. In spite of these intensive efforts, a general and decisive result on power and efficiency has completely been missing.

In this talk, we will derive universal trade-off relations between power and efficiency, and as their corollary we will give a no-go
theorem which prohibits the coexistence of finite power and the Carnot efficiency. For the case of Markovian engines, inspired by the partial entropy production [10], or the idea of decomposition of entropy production, we first derive a trade-off inequality between heat exchange and entropy production rate. Using this, we easily show the inequality between power and efficiency [11]. For the case of non-Markovian engines, with the aid of the Lieb-Robinson bound [12], we derive the inequality between the speed of operation and efficiency [13].

To reconstruct thermodynamics based on the microscopic laws is one of the most important unfulfilled goals of statistical physics. Recently, with using quantum informational techniques, many researches have tried to reconstruct and/or expand the second law of thermodynamics [1-3]. However, the following open problems remain: 1. The results are based on strong assumptions about thermodynamic systems and heat baths, e.g., the i.i.d. feature and/or the number of degeneracy. These assumptions are not necessarily satisfied by actual thermodynamic systems. 2. The analysis is mainly limited to cases isothermal case, in which the temperatures of the baths do not change. Adiabatic processes which changes temperatures of all systems radically are not clarified.

Here, we show that the first law and the second law for adiabatic processes are derived from an assumption that ``probability distributions of energy in Gibbs states satisfy large deviation, which is widely accepted as a property of thermodynamic equilibrium states [4]. We define an adiabatic transformation as a randomized energy-preserving unitary transformations on the many-body systems and the work storage. As the second law, we show that an adiabatic transformation from a set of Gibbs states to another set of Gibbs states is possible if and only if the regularized von Neumann entropy becomes large. As the first law, we show that the energy loss of the thermodynamic systems during the adiabatic transformation is stored in the work storage as "work," in the following meaning:
(i) the energy of the work storage takes certain values macroscopically, in the initial state and the final state.
(ii) the entropy of the work storage in the final state is macroscopically equal to the entropy of the initial state.
As corollaries, our results give other forms of the first and second laws, e.g., the principle of maximum work and the first law for the isothermal processes.

Biological systems make extensive use of reversible polymerization: peptides are assembled from amino-acids, actin
filaments are assembled from G-actin and glucans (carbohydrates) are assembled from monosaccharides. In this talk, inspired by a recent experimental study on the metabolism of glucans, we study the self-assembly of such polymers from the point of view of non-equilibrium thermodynamics. We first consider a closed system in which polymers dynamically evolve towards equilibrium where detailed balance is satisfied and the entropy is maximum. We then consider open systems, in which the polymers are in contact with chemostats, characterized by fixed concentrations of polymers of a given length. In accordance to a general theoretical result, we find new dynamic regimes when the number of chemostats is larger than the number of conservation laws of the chemical network. We will then discuss extensions of this framework for the self-assembly of polymers which carry information in their sequence.

The quantum steering ellipsoid formalism naturally extends the Bloch vector picture to provide a visualisation of two-qubit systems. If Alice and Bob share an entangled state then a local measurement by Bob steers Alice’s Bloch vector; given all possible measurements by Bob, the set of states to which Alice can be steered forms her steering ellipsoid inside the Bloch sphere. This gives us a novel geometric perspective on a number of quantum correlation measures such as entanglement, CHSH nonlocality and singlet fraction. In particular, by analysing a tripartite scenario we find that steering ellipsoid volumes obey a simple monogamy relation from which one can derive the well-known CKW (Coffman-Kundu-Wootters) inequality for the monogamy of entanglement. Remarkably, we can also use steering ellipsoids to derive some highly non-trivial results in classical Euclidean geometry, extending Euler's inequality for the circumradius and inradius of a triangle.

February 2016

Speaker

Prof. Jean-Charles Delvenne (Université catholique de Louvain)

Time

February 12 (Fri) 10:00-11:00

Place

工学部6号館 3F セミナー室C

Title

Entropy reduction and energy extraction in controlled systems

Abstract

We will discuss the possibility to extract energy and reduce entropy from a dynamical system thanks to feedback control, ie from the exploitation of observations on the system. In particular we will re-derive the Kalman filter from an information-theoretic perspective and discuss the impact of discrete-time dynamics, as opposed to continuous-time dynamics, on the efficiency of extraction with respect to information contained in the observation. We will also exhibit the simplest class of systems where Carnot's theorem, an open-loop statement, can be formulated and proved, leaving the possibility for feedback and finite-time extensions.

October 2015

Speaker

布能謙氏（東京大学）

Time

October 19 (Mon) 13:00-15:00

Place

工学部6号館 3F セミナー室B

Title

Work fluctuation-dissipation trade-off in heat engines

Abstract

Recent developments of nonequilibrium statistical mechanics allow us to formulate thermodynamic relations for arbitrary nonequilibrium initial and final states [1]. They can be used to quantify thermodynamic costs of information encoding and erasure processes as well as to quantify the extractable work from information heat engines. In those general situations, reducing energy dissipation allows us to increase the efficiency of a given thermodynamic task, and reducing work fluctuation allows us to prepare an exact amount of work needed to complete the task, or to extract a deterministic amount of work from the system. Thus, suppressing both work fluctuation and energy dissipation is vital to control nanosystems that work at the level of thermal fluctuations.
Previous studies have explored the regime around vanishing work fluctuations by using techniques of quantum information theory, known as the single-shot statistical mechanics [2, 3] and the regime around vanishing energy dissipation by using the fluctuation-dissipation relation and the second law of thermodynamics [1]. However, the single-shot statistical mechanics and the fluctuation-dissipation relation cannot be applied to the intermediate regime in which work fluctuation and energy dissipation take finite values. We report the trade-off relation between work fluctuation and energy dissipation for the entire regime, where the lower bound is quantified by the measure of distance between the nonequilibrium distribution and the equilibrium distribution [4]. We propose a method to construct explicit protocols that achieve the lower bound of the trade-off relation. An application of the trade-off relation to information heat engines is carried out, including a numerical simulation to test the trade-off relation.
The seminar is presented using chalk on a blackboard. Details of the proof of the trade-off relation are presented in the seminar.

In this work, we introduce stochasticity into the traditional lumping analysis, extend the lumping process from the rate equation to the chemical master equation and the stochastic differential equation, and derive the fluctuation relations between kinetically and thermodynamically equivalent networks under intrinsic and extrinsic noises. The result provides a theoretical basis for the legitimate use of low-dimensional network models in the studies of macromolecular fluctuations and related biological functions. More widely, it reveals which stochastic features different levels of contracted transition networks will or should exhibit, shedding light on the fluctuations of hierarchical networks in systems biology, chemical reactions, and general complex systems.

Speaker

Prof. Christian Van den Broeck (Universiteit Hasselt)

Time

August 24 (Mon) 10:30-11:30

Place

Faculty of Science Bldg.1, Room 913 （※理物の上田研と合同で，場所は理学部1号館です．）

Title

Onsager symmetry in periodically driven systems

Abstract

We show that — while asymmetric Onsager matrices may appear in a system under time-asymmetric periodic driving — the matrix necessarily converges to a symmetric matrix in the limit of zero dissipation. In particular, reversible efficiency can not be reached at finite power [1].
[1] Karel Proesmans & Christian Van den Broeck, arXiv:1507.00841.

Many work extraction or information erasure processes in the literature involve the raising and lowering of energy levels via external fields. But even if the actual system is treated quantum mechanically, the field is assumed to be classical and of infinite strength, hence not developing any correlations with the system or experiencing back-actions. We extend these considerations to a fully quantum mechanical treatment, by studying a spin-1/2 particle coupled to a finite-sized directional quantum reference frame, a spin-l system, which models an external field. With this concrete model together with a bosonic thermal bath, we analyse the back-action a finite-size field suffers during a quantum-mechanical work extraction process, the effect this has on the extractable work, and highlight a range of assumptions commonly made when considering such processes. The well-known semi-classical treatment of work extraction from a pure qubit predicts a maximum extractable work W = kT log 2 for a quasi-static process. We show that this holds as a strict upper bound in the fully quantum mechanical case, and is only attained in the classical limit. (arXiv:1406.3937)

Stochastic thermodynamics is a theoretical framework that assigns thermodynamic quantities -- such as work and entropy -- to individual fluctuating trajectories of small systems. As a theoretical tool, it has been useful in refining our understanding of irreversibility at the micron scale. In this talk, I develop an analogous framework for open quantum systems using the quantum trajectories formalism. By considering thermal reservoirs engineered from sequences of small quantum systems, I will be able to introduce consistent trajectory-dependent definitions of thermodynamic quantities. Furthermore, I will briefly discuss the connection between entropy production within this framework and irreversibility by way of a detailed fluctuation theorem.

Speaker

Gavin E. Crooks (Lawrence Berkeley National Laboratory)

Time

July 4 (Thu) 15:30-16:30

Place

Bldg.16, #827

Title

Molecular machines and the thermodynamic cost of nostalgia

Abstract

Molecular scale machines not only manipulate energy and matter at the nanoscale, they must also manipulate information. As a consequence, there's a tradeoff between thermodynamic efficiency, memory and prediction. A prodigious memory allows more accurate prediction of the future, which can be exploited to reduce dissipation. But the persistence of memory is a liability, since information erasure leads to increased dissipation. A thermodynamically optimal machine must balance memory versus prediction by minimizing its nostalgia, the useless information about the past [1].

Entropy production and 2nd law in stochastic systems under continuous feedback control

Abstract

Entropy production (EP) in small stochastic systems under feedback control is an issue that has attracted much theoretical attention over the last few years, at the crossroad between statistical physics and information theory [1]. In this talk, I will present some recent work in collaboration with T. Munakata (Kyoto Univ.) that focuses on systems in which measurements and actuation are performed continuously, i.e., repeated with a period shorter than the characteristic time scales of the dynamics - typically an under-damped Langevin dynamics. Two problems are investigated that correspond to actual situations:

i) the influence of measurement errors (i.e. detector noise) in a cold damping setup in which a harmonic oscillator (e.g. the cantilever of an AFM or the mirror of an interferometric detector) in contact with a heat bath is submitted to a velocity-dependent feedback force that reduces the random motion. We distinguish whether the sensor continuously measures the position of the resonator or directly its velocity (in practice, an electric current). We also assign a relaxation dynamics to the feedback mechanism and compare the apparent entropy production in the system plus the heat bath to the total entropy production in the super-system that includes the controller [2].

ii) the influence of a time delay between the input signal and the output control action, a situation that occurs in many biological or artificial systems (e.g. in the control of vision and posture, or in laser networks). We show that the system spontaneously settles into a nonequilibrium steady state where entropy is permanently produced (cooling or heating is achieved depending on the delay). However, since the feedback makes the dynamics non-Markovian, this supposes to properly revisit the definition of EP as a measure of time-irreversibility within the framework of stochastic thermodynamics [3].

In both cases, we adopt the standpoint of the controlled system and, in the spirit of [4,5], we identify the entropy pumping contribution that describes the influence of the external agent and that modifies the second law of thermodynamics and the fluctuation theorems.